L11n84

L11n83

L11n85

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n84 at Knotilus!

Link Presentations

[edit Notes on L11n84's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X9,18,10,19 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,8,18,9 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{3}-t(2)^{3}-3t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-3t(2)-t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+4q^{5/2}-5q^{3/2}+6{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-2za^{3}-a^{3}z^{-1}+2z^{3}a+2za-z^{5}a^{-1}-3z^{3}a^{-1}-3za^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+z^{6}a^{-4}+a^{4}z^{4}-3z^{4}a^{-4}+z^{2}a^{-4}-a^{4}+3z^{7}a^{-3}+2a^{3}z^{5}-11z^{5}a^{-3}-2a^{3}z^{3}+8z^{3}a^{-3}-za^{-3}+a^{3}z^{-1}+a^{2}z^{8}+3z^{8}a^{-2}-4a^{2}z^{6}-11z^{6}a^{-2}+9a^{2}z^{4}+9z^{4}a^{-2}-5a^{2}z^{2}-2z^{2}a^{-2}+az^{9}+z^{9}a^{-1}-3az^{7}+3az^{5}-10z^{5}a^{-1}+11z^{3}a^{-1}+az-2za^{-1}+4z^{8}-16z^{6}+20z^{4}-8z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               2 1   -1 4             3 2     1 2           3 2       -1 0         3 3         0 -2       3 4           1 -4     1 2             -1 -6     3               3 -8 1 1                 0 -10 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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